Modelling and interpreting the coupling between coherent nonlinear structures and ambient turbulence in fusion plasmas using approaches derived from the Lotka Volterra equations

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Modelling and interpreting the coupling between

coherent nonlinear structures and ambient

turbulence in fusion plasmas using approaches

derived from the Lotka-Volterra equations

by

Hao Zhu

Thesis

Submitted to the University of Warwick for the degree of

Doctor of Philosophy

Department of Physics

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List of Tables

2.1 The physical meaning of the terms and parameters in the r.h.s of the

MD model. . . 68

2.2 The physical meaning of the terms and parameters in the r.h.s of the II model. . . 73

3.1 The physical meanings of terms and parameters in r.h.s of ZCD model equations(3.1.1-3.1.4). . . 79

3.2 Summary of Figs.3.2.1 to 3.3.5 . . . 90

3.3 Properties of stability analysis of MD system . . . 94

3.4 Properties of stability analysis of ZCD system . . . 95

5.1 Experimentally inferred parameter values[Dendy et al., 2013] for both Te rise andTedrop cases. We have∂κT/∂x1 =∂κT/∂x3,∂κQ/∂x1= ∂κQ/∂x3. . . 126

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List of Figures

1.2.1 Fig.3.4 in [Chen, 1975], the diamagnetic drifts in a cylindrical plasma 7 1.2.2 Fig.2(a) in [Yushmanov et al., 1990], the comparisons of confinement

times(τ_EIT ER89−P and τ_Eexp) in different fusion devices. . . 14 1.2.3 Fig.7 in [Doyle et al., 2007], the comparison of experimental power

threshold for L-H transition with the scaling expression Eq.(1.2.81)[Wesson, 2011]. . . 15 1.2.4 Fig.3 in [Cheng et al., 2013], the transitions of L-I-H and L-I. . . 16 1.2.5 Fig.6-13 in [Chen, 1975], the geometry of drift instability in a cylinder.

The rectangular region is also shown in Fig.1.2.6. . . 18 1.2.6 Fig.6-14 in [Chen, 1975], the physical mechanism of drift waves. . . . 19 1.2.7 Fig.3.3 in [Horton et al., 2012], the diagram of the Carnot cycle for

∇T driven drift waves. W is the maximum energy released to the plasma turbulence. . . 20 1.2.8 Fig.18.2 in [Itoh et al., 1999], the effect of a sheared E ×B flow,

VE×B on a turbulent eddy. (a) illustrates the Cartesian coordinate

with magnetic field and shear velocity. (b) the circular turbulent eddy with the sizeL. (c) distorted turbulent eddy by sheared flow. . . 21 1.2.9 Fig.1 in [Diamond et al., 2005], classic and new paradigm for plasma

turbulence. . . 22 1.2.10Fig.1 in [Mantica et al., 1999], the time evolution of electron

temper-ature Te, the averaged electron density ¯ne and the electron energy

stored in the plasmas We for RTP discharge r19970224.024. A

hy-drogen pellet is injected att= 0.2054 second in target plasmas. . . . 23 1.2.11Fig.1 in [Inagaki et al., 2010], typical electron temperature Te

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1.2.12Fig.2 in [Inagaki et al., 2010], the bifurcation diagram containing stationary and dynamic state. This is expressed by the relation-ship between heat flux average by electron densityqe/neand electron

temperature gradient∇Te in the core plasma(ρ= 0.19) in LHD. The

green arrows denote the variation directions. . . 25 1.3.1 Schematic diagram of a Tokamak device from Wikipedia. . . 26 1.3.2 Figure 1.6.3 in [Wesson, 2011]. (a) The change of flux through the

torus induces toroidal electric field which drives the toroidal current. (b) The flux change is produced by primary winding using a trans-former core. . . 27 1.3.3 Figure 5.2 in [Freidberg, 2007]. The cross section of a Tokamak. . . . 28 1.3.4 Fig.12.3.1 in [Wesson, 2011]. The configuration of the Joint European

Torus(JET) Tokamak. . . 29 1.3.5 Schematic diagram of structure of Stellarator from the Wikipedia. . 31 1.3.6 Comparison of Tokamak and Stellarator from the Wikipedia. . . 31 1.3.7 Schematic diagram of Large Helical Device(LHD). . . 32 1.4.1 Fig.7.7.2 in [Wesson, 2011]. Four phases of plasma disruption. . . 34 1.4.2 Fig.7.6.1 in [Wesson, 2011]. The X-ray emission from (a) the central

region (b) the outer region of plasma. . . 34 1.4.3 Schematic diagram of tearing mode in a poloidal cross section of a

Tokamak from Richard Fitzpatrick’s report. . . 35 1.4.4 Fig.6.13.1 in [Wesson, 2011]. Showing the destabilizing curvature on

the outer side of the Tokamak and stabilizing on the inner side. . . . 36 1.4.5 Fig.7.17.1 in [Wesson, 2011]. This figure demonstrates the falling in

temperature, density and pressure respectively resulting from ELMs in JET. . . 36 2.2.1 2-ODE Lotka-Volterra model. Defining R as rabbits, F as foxes.

Left panel: time series of 2-ODE Lotka-Volterra model. Right panel: phase plot of 2-ODE Lotka-Volterra model. The parameters and initial conditions are given as α = 1, β = 0.1, γ = 2, δ = 0.15, R0 = 35, F0 = 40. . . 45 2.2.2 Pendulum oscillation model with small angle approximation. Left

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2.2.3 Pendulum oscillation model with large angle. Left panel: time series of pendulum oscillation model. Right panel: phase plot of pendulum oscillation model. The parameter and initial conditions are A = 2, x0= 0, v0 = 2.8. . . 47

2.2.4 Pendulum oscillation model in critical angle condition. Left panel: time series of pendulum oscillation model. Right panel: phase plot of pendulum oscillation model. The parameter and initial conditions areA= 2, x0 = 0, v0 = 2.8285. . . 48 2.2.5 Figure 7.0.1 in [Strogatz, 2014]. Three types of limit cycles, which

are stable limit cycle, unstable limit cycle and half-stable limit cycle. 49 2.2.6 Van der Pol oscillation model. Left Panel: time series of Van der Pol

oscillation model. Right Panel: phase plot of Van der Pol oscillation model. The parameter and initial conditions areη= 2, x0= 0, v0 = 1.2. 50 2.2.7 Van der Pol oscillation model. Left panel: time series of Van der Pol

oscillation model. Right panel: phase plot of Van der Pol oscillation model. The parameter and initial conditions areη= 8, x0= 0, v0 = 1.2. 50 2.2.8 Figure 7.1.4 and Figure 7.1.5 in [Strogatz, 2014]. Numerical solution

of Van der Pol equation forη = 1.5 and starting from ( ˙x, x) = (0.5,0) att= 0. . . 51 2.2.9 Figure 1.1 in [Hilborn, 1994]. The inductor-diode circuit. i(t) is the

electric current. Vd(t) is the electric potential difference across the

diode. . . 52 2.2.10Figure 1.3 in [Hilborn, 1994]. The time series of diode voltage. Upper

panel shows period-1, while lower panel illustrates period-2. . . 52 2.2.11Figure 1.4 in [Hilborn, 1994]. The time series of diode voltage. Upper

panel shows period-4, while lower panel indicates period-8. . . 53 2.2.12Figure 1.5 in [Hilborn, 1994]. Upper panel is period-4 time series of

diode voltage. Lower panel is the corresponding time series of current. 53 2.2.13Figure 1.6 in [Hilborn, 1994]. Diode voltage as a function of time.

Both of upper and lower panels are no longer periodic. . . 54 2.2.14Figure 1.7 in [Hilborn, 1994]. Upper panel is period-3 time series

of diode voltage. Lower panel is the signal generator voltage as a function of time. . . 54 2.2.15Figure 1.8 in [Hilborn, 1994]. Bifurcation diagram for the diode circuit. 55 2.2.16Figure 1.9 in [Hilborn, 1994]. Another bifurcation diagram for a

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2.2.17Lorenz chaotic model. Left panel: irregular time series of Lorenz chaotic model. Right panel: phase plot of Lorenz chaotic model in the type of butterfly. The parameters and initial conditions areσ = 10, b= 8

3, r= 28, x0 =z0 = 0, y0 = 1. . . 57

2.2.18Reconstruction of phase plot of Lorenz attractor with Takens’ Theorem. 58 2.3.1 Deviations of the constant of motion in Eq.(2.2.18) with various fixed step sizes. . . 61

2.3.2 Bifurcation diagram in Chapter 4.4 with step sizeδt= 0.5 . . . 61

2.3.5 Bifurcation diagram in Chapter 4.4 with step size δt = 0.001. This figure is from Fig.4.6.6 in Chapter 4.4. . . 63

2.3.6 Fig.17.1.3 in [Press, 2007]. For fourth-order Runge-Kutta method, in each step the derivative is evaluated four times. . . 64

2.3.7 Fig.17.3.1 in [Press, 2007]. This is extrapolation used in the Bulirsch-Stoer method. A large intervalH is spanned by various sequences of finer and finer substeps. . . 65

2.4.1 Fig.2 in [Malkov and Diamond, 2009]. Upper panel indicates time evolution ofE(normalised micro-scale turbulence intensity) andN(normalised electron temperature gradient). Lower panel demonstrates the time series ofU(energy of zonal flow). The system starts from overpowered H-mode then unstable L-mode then transient oscillatory T-mode. . . 69

2.4.2 Fig.3 in [Malkov and Diamond, 2009]. Limit cycle oscillation of E(normalised micro-scale turbulence intensity), N(normalised elec-tron temperature gradient) andU(energy of zonal flow) whenq= 0.58. 70 2.4.3 Fig.4 in [Malkov and Diamond, 2009]. Collapse of limit cycle oscil-lation and transition to quiescent H-mode(QH-mode), in which only N(normalised electron temperature gradient) exists whenq = 0.582. 70 2.4.4 Fig.6 in [Malkov and Diamond, 2009]. The external heating rate slowly varies from q = 0.47(stable L-mode) to q = 0.62(stable QH-mode) and back. It is found that the final state is not identical with the initial state, which is identified as hysteresis of the dynamical system. . . 71

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2.5.1 Fig.1 in [Dendy et al., 2013], time series of normalised (a)δ∇Te, (b)δq

and (c)δTe for the Te rise case in LHD in ρ = 0.19. Blue lines are

experimental data and red lines are model outputs. Parameters are:

χ0 = 3.2m2s−1, Te0 = 3.5keV, Lc = 1.1m, κT0 = 15, ∂κT/∂x1 = ∂κT/∂x3 = 1.5, κQ0= 225, ∂κQ/∂x1 =∂κQ/∂x3 = 22.5, γL1 =γL2 =

35 andη/τcχ0 = 10.5. . . 75

2.5.2 Fig.2 in [Dendy et al., 2013], time series of normalised (a)δ∇Te, (b)δq

and (c)δTe for theTe drop case in LHD in ρ = 0.19. Blue lines are

experimental data and red lines are model outputs. Parameters are:

χ0 = 2.4m2s−1, Te0 = 2.9keV, Lc = 1.1m, κT0 = 20, ∂κT/∂x1 = ∂κT/∂x3 = 2.0, κQ0 = 400, ∂κQ/∂x1 = ∂κQ/∂x3 = 40, γL1 =γL2 =

35 andη/τcχ0 = 10.5. . . 76

3.2.1 Upper panel: From a state near overpowered H-mode to unstable H-mode then to unstable L-mode then to T-mode. Lower panel: Transition to T-mode for U1 and U2 showing intersection at t'750

time units followed by gradual energy reversal. The model parameters areν1 = 19,ν2 = 1.01ν1, η1 = 0.12, η2 = 1.01η1, q = 0.47, ρ= 0.55, σ= 0.6,ζ = 1.7. . . 82 3.2.2 Upper panel: Transition from unstable fixed point state(T-mode)

to unstable limit cycle oscillation state. Lower panel: Zoomed in version fromt= 300 tot= 800. The model parameters are ν1 = 19, ν2 = 0.01ν1,η1 = 0.12,η2 = 0.1η1,q= 0.47, ρ= 0.55,σ = 0.6,ζ = 1.7. 83 3.2.3 Burst energy transfer from U2 toU1 during strong nonlinear

oscilla-tion, followed by limit cycle oscillation in N,E and U1. The model

parameters areν1 = 19,ν2 = 1.01ν1,η1 = 0.12,η2 = 1.01η1,q= 0.58,

ρ= 0.55,σ = 0.6,ζ = 1.7. . . 83 3.2.4 Upper panel: Collapse of limit cycle in N, E and U1. Lower panel:

Stair increasing ofU2 between every two pulses. The model param-eters are ν1 = 19, ν2 = 0.01ν1, η1 = 0.12, η2 = 0.01η1, q = 0.58, ρ= 0.55,σ = 0.6,ζ = 1.7. . . 84 3.2.5 Upper panel: Collapse of limit cycle with positively correlated growth

of pulses ofU1andU2. Lower panel: Zoomed in version fromt= 240

to t = 400. The model parameters are ν1 = 19, ν2 = 1.0001ν1,

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3.2.6 Evolution to the finite N attractor for different η2 values. Upper

panel: η2 = 0.05. Middle upper panel: η2 = 0.06. Middle lower panel: η2= 0.10. Lower panel: η2= 0.11. The remaining parameters

are identical: ν1 = 19, ν2 = 1.001ν1,η1 = 0.12, q = 0.582, ρ= 0.55, σ= 0.6,ζ = 1.7. . . 85 3.3.1 First panel: Fig.2 in MD. The parameters are ν = 19, η = 0.12,

q = 0.47, ρ = 0.55, σ = 0.6, ζ = 1.7. Second panel: Projection of first panel onE-U plane. Third panel: Phase plot of Fig.3.2.1. Last panel: Phase plot of Fig.3.2.1 with 81 initial conditions. Stars denote initial values, blue dots denote trajectories and red diamonds denote final states. . . 87 3.3.2 First panel: Projection of Fig.2 in MD onE-U plane. The parameters

are ν = 19, η = 0.12, q = 0.47, ρ = 0.55, σ = 0.6, ζ = 1.7. Second panel: Phase plot of Fig.3.2.2. Last panel: Phase plot of Fig.3.2.2 with 81 initial conditions. Stars denote initial values, blue dots denote trajectories and red diamonds denote final states. . . 88 3.3.3 First panel: Fig.3 in MD. The parameters are ν = 19, η = 0.12,

q = 0.58, ρ = 0.55, σ = 0.6, ζ = 1.7. Second panel: Projection of first panel onE-U plane. Third panel: Phase plot of Fig.3.2.3. Last panel: Phase plot of Fig.3.2.3 with 81 initial conditions. Stars denote initial values, blue dots denote trajectories and red diamonds denote final states. . . 88 3.3.4 First panel: Projection of Fig.3 in MD onE-U plane. The parameters

are ν = 19, η = 0.12, q = 0.58, ρ = 0.55, σ = 0.6, ζ = 1.7. Middle panel: Phase plot of Fig.3.2.4. Last panel: Phase plot of Fig.3.2.4 with 81 initial conditions. Stars denote initial values, blue dots denote trajectories and red diamonds denote final states. . . 89 3.3.5 First panel: Phase plot for Fig.4 of MD. Second panel: Projection

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3.5.3 Parameter space of ν2/ν1 and η2/η1 when q= 0.582. . . 97

4.2.1 KD/MD model dependence of ratios of confinement time τQH/τT

(blue stars; left scale) and temperature gradientNQH/NT (red crosses;

right scale) on the normalised increase in heating rate δq/q0. Solid line for τQH/τT is inferred from Eqs.(4.2.5) and (4.2.6). Points are

obtained from numerical results forδq = 0.495,τQH = 1.8182,q0

val-ues are shown in figure; other parameter valval-ues areν= 19, η= 0.12,

ρ= 0.55,σ = 0.6,ζ = 1.7. . . 103 4.2.2 ZCD model dependence of ratios of confinement time τQH/τO (blue

stars; left scale) and temperature gradient NQH/NO (red crosses;

right scale) on the normalised increase in heating rate δq/q0. Solid line is inferred from Eqs.(4.2.4) and (4.2.9). Points are obtained from numerical results for δq = 0.20, τQH = 1.8182, q0 values are shown

in figure; other parameter values are ν1 = 19, ν2 = 0.19, η1 = 0.12,

η2 = 0.012,ρ= 0.55,σ = 0.6,ζ = 1.7. . . 104

4.3.1 Time traces and phase space evolution for the KD/MD model, Eqs(4.1.1) to (4.1.3), with a discontinuous increase in heating rateq by amount

δq = 0.16 from q0 = 0.45 at t = 2000 time units; q reverts to q0 at t= 3000 time units. Upper left plot shows time traces of variablesN

(black),q (dashed magenta), U (red) and E (green). Lower left plot shows time trace of energy confinement timeτcdefined by Eq.(4.2.1).

Right plot shows time evolution of the system in (N, E, U) phase space. The sequence of key phases is labelled in all three plots in this Figure as follows. A is the initial transient evolution from the over-powered H-mode point I, leading to convergent cyclic motion towards fixed point attractor B corresponding to T-mode. At C the instantaneous increase in heating rateq induces rapid departure from the T-mode attractor B to the QH-mode (increasedN; E =U = 0) attractor D with improved confinement time. Instantaneous rever-sion of q to initial value q0 brings the end of phase D and results in immediate transition to a QH-mode by exponential decrease in N, labelled E, with a lower value of N and the same confinement time as phase D. There is later a spontaneous back transition from E at

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4.3.2 As Fig.4.3.1, for the case whereq0 = 0.47. The major difference is the

longer duration of the post-heating QH-mode phase E, after reversion ofq to its initial value. . . 107 4.3.3 As Fig.4.3.2, for the case where q0 = 0.49. The major difference is

that the back transition from the post-heating QH-mode phase E, which is not a stable attractor, has not yet occurred by the end of this run. . . 108 4.4.1 As Fig.4.3.1, for the two-predator ZCD model, Eqs(4.1.4) to (4.1.7),

with a sharp heating transition where q0 = 0.45,δq= 0.16. The

sec-ond predator fieldU2 is traced in blue in the upper left plot. The ma-jor difference from Fig.4.3.1 is that the post-heating T-mode state F is a repulsive fixed point, from which the system spontaneously tran-sitions and converges cyclically to the fixed point G. This is known from [Zhu et al., 2013] and has enhancedN and finiteU2, withEvery small. Here we refer to the attractive fixed point G as an example of O-mode. . . 109 4.4.2 As Fig.4.3.2, for the two-predator ZCD model[Zhu et al., 2013] with

a sharp heating transition where q0 = 0.47, δq = 0.16. The major

difference from Fig.4.3.2 is that the post-heating T-mode state F is a repulsive fixed point, from which the system spontaneously tran-sitions and converges cyclically to the limit cycle G. This is known from [Zhu et al., 2013] and has oscillations of enhancedN and finite

U2, accompanied by small pulses ofE. Here we refer to the attractive

limit cycle G as an example of O-mode. . . 110 4.4.3 As Fig.4.3.3, for the two-predator ZCD model[Zhu et al., 2013] with

a sharp heating transition where q0 = 0.49, δq = 0.16. There is

insufficient run time for the phase E QH-mode to transition to T-mode and then to the O-T-mode attractor, unlike Figs.4.4.1 and 4.4.2. 111 4.5.1 Time traces and phase space evolution for the MD model, Eqs(4.1.1)

to (4.1.3), with a smooth increase, represented by the tanh function in Eqs.(4.5.1) and (4.5.2), in the heating rateq by an amountδq = 0.16 from q0 = 0.45 around t = 2000 time units; q reverts to q0 around t= 3000 time units. Upper left plot shows time traces of variablesN

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4.5.2 As Fig.4.5.1, for the case where q0 = 0.49. There are two major

differences from Fig.4.3.3. First, confinement time τc during phase

C experiences a transient drop before jumping to higher confinement regime. Second, a spontaneous back-transition F to T-mode appears after long duration phase E QH-mode. . . 113 4.6.1 Period-1 oscillation in ZCD system dynamics in response to the

vary-ing heatvary-ing rate defined by Eq.(4.6.1) withA= 0.0215; other param-eter values are as for Fig.4.4.2. Left, the power spectrum ofN; right, the full attractor in (N, U2, E) space; inset, the time series of N. . . 115

4.6.2 As Fig.4.6.1, showing period-2 oscillation in ZCD system dynamics when A= 0.0240. . . 115 4.6.3 As Fig.4.6.1, showing period-4 oscillation in ZCD system dynamics

when A= 0.0270. . . 116 4.6.4 Period-doubling illustrated by the power spectra ofN from Figs.4.6.1

to 4.6.3, over-plotted in the frequency range from 0.04 to 0.08. Blue, red and black dash lines denote spectra of period-1, period-2 and period-4 respectively. . . 116 4.6.5 As Fig.4.6.1, showing chaotic attractor of the ZCD system dynamics

when A= 0.0295. . . 117 4.6.6 Period-doubling bifurcation path to chaos of ZCD system dynamics.

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5.3.1 Time evolution of electron temperature at multiple radial locations, derived from LHD data(blue) and the model(red) for the core tem-perature rise(R) heat pulse propagation experiment in plasma 49708. Radial locations range from edge(ρ = 0.703) to core(ρ = 0.015), where ρ = r/a, r is the radial co-ordinate and a ∼ 0.6m is minor radius of LHD. Model results match experimental data well from

ρ= 0.450 inwards to the plasma core, especially amplitudes and the time structure of pulse decay. The amplitudes of model time traces increase from edge to core, as in the measured electron temperature profiles. Model results do not fit experimental data outwards from

ρ= 0.546 to ρ= 0.703, implying that different physics applies in the outer LHD plasma. . . 127 5.3.2 Time evolution of electron temperature at multiple radial locations,

derived from LHD data(blue) and the model(red) for the core temper-ature drop(D) heat pulse propagation experiment in plasma 49719. Radial locations range from edge(ρ = 0.703) to core(ρ = 0.015), whereρ=r/a,r is the radial co-ordinate anda∼0.6m is minor ra-dius of LHD. As in Fig.5.3.1, model results match experimental data well fromρ= 0.450 inwards to the plasma core. Again, model results do not fit experimental data outwards from ρ = 0.546 to ρ = 0.703, reinforcing that different physics dominates in the outer LHD plasma. 128 5.3.3 Time evolution of electron temperature at three specific radii selected

from Fig.5.3.1 for the central temperature rise(R) case, during the heat pulse propagation experiment in LHD plasma #Te49708. Data and model output are denoted by blue and red lines respectively. . . 128 5.3.4 Time evolution of electron temperature at three specific radii selected

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5.3.5 Data analysis underpinning calculation of pulse velocity from the ex-perimental data, which requires a statistically robust identification of the time of the pulse peak from the noisy data at each radius 0.450≥ ρ ≥ 0.015. Blue lines show timeseries of electron tempera-ture data versus time from the R case(Te49708). Red lines denote timeseries smoothed over a window whose span is 5% of the total sample points, so that approximately 50 sample points generate the moving average. Black dots mark the maximum values, at each ra-dius, of each smoothed time-evolving electron temperature pulse. The width of the horizontal error bars is defined by span of the moving window. The black dash line is the best fit straight line joining all the peaks. From it we infer the pulse propagation speed, which is nearly independent of radius, to be (32.62±9.89)ms−1. . . 130 5.3.6 As Fig.5.3.5, demonstrating the calculation of the pulse velocity for

the electron temperature data for the D case. Despite data which is more noisy than in Fig.5.3.5, the pulse propagation velocity calculated from the dashed line is approximately constant across all radii in this region 0.450 ≥ ρ ≥ 0.015 of LHD plasma 49719. We infer a pulse velocity (53.50±20.97)ms−1. . . 131 5.3.7 Variation and robustness of nonlinear pulse phenomenology in the

model, for three different values of pseudo-velocity v0 at three radial

locations(Left panel). Blue lines show experimental data for R case in plasma 49708. Solid, dash and dot magenta lines denote simulation outputs forv0= 15, v0 = 30 andv0 = 45 respectively. The boundary

condition for R case isy2(0) = 1.5, and no horizontal or vertical shift is applied to the model outputs. Right panel: Phase plot of model outputs from left plots, all of which lie on the same orbit. Circulation direction of this phase plot is identical with Fig.6(a) of [Dendy et al., 2013], where the sign of the horizontal axis is reversed. . . 131 5.3.8 Impact of different ξ values on the heat pulse model in R case, four

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Acknowledgments

I would like to thank my supervisors Professor Sandra C Chapman and Professor Richard O Dendy for their support over the three and half years. I am grateful for helpful discussions with Professor Shigeru Inagaki, Professor Sanae-I. Itoh, Professor Kimitaka Itoh and Professor George Rowlands. Moreover, of course, I will also be thankful for the unwavering support from my parents.

Chapter 3 was part-funded by the EPSRC and the RCUK Energy Programme under grant EP/I501045 and the European Communities under the contract of As-sociation between EURATOM and CCFE.

Chapter 4 was part-funded by the support by KAKENHI (21224014, 23244113) from JSPS and the EPSRC and the RCUK Energy Programme under grant EP/I501045 and the European Communities under the contract of Association between EU-RATOM and CCFE.

Chapter 5 was part-funded by the RCUK Energy Programme under grant EP/I501045. This project has received funding from the Euratom under grant agreement number 633053. This work was also supported in part by the grant-in-aid for scientific research of JSPF, Japan (23244113) and KAKENHI (21224014) from JSPS, Japan.

This thesis was typeset with LA_{TEX 2}_ε1 _{by the author.}

1_LA_{TEX 2ε}_{is an extension of L}A_{TEX. L}A_{TEX is a collection of macros for TEX. TEX is a trademark}

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Declarations

I hereby declare that this thesis is my own work, except where explicitly stated, and that it has not been submitted for another degree at the University of Warwick, or any other university.

Chapters 1 and 2 do not include original work but provide theoretical and experimental backgrounds. The results presented in Chapters 3 and 4 have been published in [Zhu et al., 2013] and [Zhu et al., 2014] respectively. Moreover, results in Chapter 5 have been submitted to [Zhu et al., 2015].

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Abstract

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Summary

When facing the energy crisis, it is recognised worldwide that nuclear fusion is one of the most promising potential solutions. However, plenty of unexplored topics still exist in fusion plasma physics. This thesis focuses primarily on one of those issues: modelling and interpreting the coupling relationships between coherent nonlinear structures and ambient micro-scale turbulence in fusion plasmas, using approaches derived from the Lotka-Volterra(or predator-prey) model. Additionally, we interpret heat pulse propagation experiments in the Large Helical Device(LHD), which is a type of fusion device called Stellarator, in terms of a new travelling wave transformation of an existing model. Introductions to fundamental plasma physics, L-H transitions, drift wave turbulence–zonal flows interactions, heat pulse propagation experiments, MHD description and instabilities in Tokamaks are given inChapter 1. InChapter 2, we introduce some key aspects of nonlinear dynamics that are helpful in analysing those phenomena in nuclear fusion. Terminologies of nonlinear dynamics such as the Lotka-Volterra(or predator-prey) model, limit cycle manifold, bifurcation theory and chaos will be described. The specific methodology applied throughout this thesis will be introduced as well.

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con-finement and of transitions between concon-finement regimes. A prime zero-dimensional paradigm is Lotka-Volterra or predator-prey. Within this framework, we propose a novel model. This requires a fourth coupled nonlinear ordinary differential equation, together with a fourth variable which we treat as geodesic acoustic modes(GAMs). We investigate the degree of invariance of the phenomenology generated by the model of Malkov et al.(3-ODE), given this additional physics. We study and com-pare the long-time behaviour of the three-equation and four-equation systems, their evolution towards the final state, and their attractive fixed points and limit cycles. We explore the sensitivities of paths to attractors. It is demonstrated that an attrac-tive fixed point of the three-equation system can become a limit cycle manifold of the four-equation system. Addressing these questions is particularly important for models that generate sharp transitions in the values of system variables which may replicate some key features of confinement transitions. Our results help to estab-lish the robustness of the zero-dimensional model approach to capturing observed confinement phenomenology in Tokamak fusion plasmas. We report these results in

Chapter 3, see also [Zhu et al., 2013].

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It is observed that rapid edge cooling of magnetically confined plasmas, in-duced by pellet injection, can trigger heat pulses that propagate rapidly inward. These can result in either large positive or negative deviations of the electron tem-perature at the plasma core. They represent a fairly extreme example of coherent nonlinear structures in plasmas, and as such are a particularly interesting challenge to theory. By applying a travelling wave transformations, we extend the model of Dendyet al., which successfully describes local temporal evolution in these plasmas, to include also spatial dependence. The extended model comprises two coupled nonlinear first order ordinary differential equations for the (x, t) evolution of the deviation from steady state of two independent variables: the excess electron tem-perature gradient and the excess heat flux, both of which are measured in the LHD experiments. Pulse velocity is also defined in terms of plasma quantities. This en-ables us to model spatio-temporal pulse evolution in a way which yields as output the spatiotemporal evolution of the electron temperature, which is also measured in detail in the experiments. We compare the model results against LHD datasets using appropriate initial and boundary conditions. Sensitivities of this nonlinear model with respect to plasma parameters, initial conditions and boundary condi-tions are investigated. We conclude that this model can match experimental data for the time-evolving temperature profiles of pulses and their propagation velocities across a broad radial range from plasma edge to core. We report these results in

Chapter 5, see also [Zhu et al., 2015].

We summarize in Chapter 6 the ways in which our new models can suc-cessfully assist interpretation of the drift wave turbulence – zonal flows interactions and the heat pulse propagation experiments. These results appear to reinforce the validity of Lotka-Volterra models when modelling and interpreting this class of phe-nomenon observed in nuclear fusion plasmas.

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this thesis. In Chapter 3, the reduced models to interpret the interactions be-tween drift wave turbulence and zonal flows will be presented. In Chapter 4, we explore the impact of different levels of external heating flux on the reduced models.

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Chapter 1

Introduction to fusion plasma

physics

1.1 Lawson criterion

In order to reach the condition of nuclear fusion ignition, we have to satisfy the triple product, which is called the Lawson criterion[Lawson, 1955]:

nT τE >5×1021m−3keV s (1.1.1)

where n and T are the peak density and the peak temperature of the particles in the plasma, andτE is the confinement time of energy[Wesson, 2011].

Under this circumstance, the plasma will have achieved the power balance. The balance equation is

PH +

1 4n

2 _{< σv >}_E αV =

3nT τE

V (1.1.2)

where PH is the external heating power, n is the particle density, σ is the

cross-section of the reaction,< σv > is the rate of nuclear reaction, E_α is the α-particle heating per unit volume, V is the plasma volume, T is the temperature and τE

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are utilized to interpret the transition to H-mode, please see Chapter 2.4.

Since such a high temperature is incompatible with confinement by material walls, another way of confinement is required. Magnetic confinement fusion(MCF) provides such a method, in which plasmas are confined in a toroidal region by the magnetic field, being held by the field in microscale gyrating orbit[Wesson, 2011]. For further information about magnetic confinement fusion devices like Tokamak and Stellarator, please see Chapter 1.3.

1.2 Fundamental plasma physics

1.2.1 Fluid description

The characteristic density of a plasma in medium-size MCF experiment would be 1019m−3[Dendy, 1990]. If each particle has a complex trajectory, it would be rather difficult to predict the behaviour of the plasma. Fortunately, the majority of the plasma phenomena observed, for example the drift wave in Chapter 1.2.5, can be described by a simple model. This model is utilized in fluid mechanics, and only the motion of the fluid elements is taken into consideration. The properties of each in-dividual particle are implicitly summed and averaged. The only difference between the ordinary fluid description and the plasma fluid description is charged parti-cles[Chen, 1975]. In this subchapter, the fundamental fluid description of plasma will be given. The fluid drift perpendicular to the magnetic field, which is not only a valid approximation of the fluid description but also linked to many fusion plasma phenomena, will be introduced as well. The discussions in the Chapter 1.2.1 appear in [Chen, 1975] and [Krall and Trivelpiece, 1986].

The equation of motion for a single particle is:

mdv

dt =q(E+v×B) (1.2.1)

Assume there are no thermal motions and no collisions. Then all particles in a fluid element move together, and the individual particle velocityv is identical with the average velocity u of the particles in the fluid element. Then the fluid equation is obtained by multiplying the density of particlesn.

mndu

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In Eq.(1.2.1), the time derivative is to be taken at the position of the parti-cles(called Langrange method). However, the equation for fluid elements relative to a co-ordinate system fixed in space(called Euler method) is more convenient for the present description.

To make this transformation to variables in the fixed frame, considerG(x, t)

to be any physical quantity of a fluid element in one-dimensional space. The variance ofG(x, t)with time in a frame moving with the fluid is:

dG(x, t)

dt = ∂G

∂t + ∂G

∂x dx

dt = ∂G

∂t +ux ∂G

∂x (1.2.3)

In three-dimensional case, Eq.(1.2.3) has been changed as:

dG

dt = ∂G

∂t + (u· ∇)G (1.2.4)

This equation is called convective derivative andu· ∇ is a scalar differential operator. In the plasma, we assume that the fluid velocity isu, then we can rewrite Eq.(1.2.2) as:

mn

∂u

∂t + (u· ∇)u

=qn(E+u×B) (1.2.5)

The complete statistical description of a given system ofN particles is given in terms of a distribution function

F = F(x1,x2, ...,xN,v1,v2, ...,vN, t) (1.2.6)

whereR

F dx1, dx2, ..., dxN, dv1, dv2, ..., dvN = 1.

The many-body distribution function Eq.(1.2.6) obeys the Liouville equation

∂F ∂t +

X

i

∂F ∂xi

·vi+ ∂F ∂vi

·aT_i

= 0 (1.2.7)

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∂ ∂tf

(1) α +v1·

∂ ∂x1

f_α(1)+ qα

mα

hE+v1×Bi ·

∂ ∂v1

f_α(1) = ∂f

(1) α ∂t c (1.2.8)

whereE andB are electric and magnetic fields respectively,fα(1) is the distribution

function of particle of type α at x1 and v1. The equation above is Boltzmann

equation. If we neglect the collisional term on the right-hand side, the equation becomes Vlasov equation.

The integral of Eq.(1.2.8) over all velocity space is

Z

∂

∂tfα+v· ∂ ∂xfα+

qα mα

hE+v×Bi · ∂ ∂vfα

dv =

Z _∂f α ∂t c

dv (1.2.9)

wherefα≡fα(x,v, t). The Eq.(1.2.9) is called equation of continuity.

The integral over all velocity space of the product of the plasma kinetic equation Eq.(1.2.8) and the momentummαv of a particle of speciesα is

Z

mαv

∂fα ∂t +v·

∂fα ∂x +

qα mα

hE+v×Bi ·∂fα ∂v

dv =

Z

mαv ∂fα ∂t c

dv(1.2.10)

With the aid of equation of continuity, Eq.(1.2.10) reduces to the momentum transfer equation for particles of speciesα; that is,

nαmα ∂Vα

∂t +nαmαVα· ∇Vα−nαqαhE+v×Bi+∇ ·Pα

=−X β

nαmα(Vα−Vβ)hναβi

(1.2.11)

wherehναβiis a mean collision frequency for momentum transfer from all other types

of plasma particles[Krall and Trivelpiece, 1986]. This equation has an alternative form as given below,

mn

∂u

∂t + (u· ∇)u

=qn(E+u×B)− ∇ ·P −mn(u−u0)

τ (1.2.12)

whereP is defined as stress tensor(∇ ·P =∇p in most of the cases, wherep is the pressure), u0 is the velocity of the neutral fluid, u−u0 is the relative velocity in

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We also have the equation of continuity

∂n

∂t +∇ ·(nu) = 0 (1.2.13)

A conservation of energy equation is also required. This is represented by the equation of state,

p=Cργ (1.2.14)

where C is a constant and γ is the ratio of specific heats Cp/Cv. The term ∇p is

then given by

∇p p =γ

∇n

n (1.2.15)

wherep=nKT. Hence, we consider the ions and electrons in plasma fluid descrip-tion. Then the charges and the current densities are given by

σ=niqi+neqe (1.2.16)

j =niqivi+neqeve (1.2.17)

Since the single particle motion will not be considered, we then usev instead of u for the velocity of fluid. We also neglect the terms representing the viscosity and collisions. Thus we get the complete set of fluid equations in this plasma fluid description.

0∇ ·E = niqi+neqe (1.2.18) ∇ ×E = −B˙ (1.2.19)

∇ ·B = 0 (1.2.20)

µ−₀1∇ ×B = niqivi+neqeve+0E˙ (1.2.21) mjnj

∂vj

∂t + (vj · ∇)vj

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∂nj

∂t +∇ ·(njvj) = 0 (1.2.23) pj = Cjn

γj

j (1.2.24)

where j = i, e are the particle species of each fluid considered, n is the particle density,p is the pressure,v is the velocity,E is the electric field,B is the magnetic field,0 and µ0 are the permittivity and permeability in vacuum respectively. The

simultaneous solution of equations above provides a set of motions and correspond-ing fields in the plasma fluid approximations[Chen, 1975]. The discussions above can be found in [Chen, 1975] and [Krall and Trivelpiece, 1986]

Fluid drift perpendicular to B

In this subchapter, the fluid with drifts perpendicular to the magnetic fieldB will be discussed. This phenomenon also appears in the generation of drift waves which will be introduced in Chapter 1.2.5. The suppression of drift wave turbulence in a fusion device may give rise to the transition to the high confinement mode(H-mode)[Diamond et al., 2005], which is mentioned in Chapter 1.1 and will be discussed in 1.2.4 and 1.2.5 in detail. The theory of fluid drifts, particularly perpendicular to the magnetic field, is one of the most important topics underpinning the transport phenomena for energy and particles in fusion plasmas. The discussions in this subchapter appear in [Chen, 1975].

We start the derivation from the equation of motion.

mn

∂v

∂t + (v· ∇)v

= qn(E+v×B)− ∇p (1.2.25)

Consider the ratio of the first term on the left-hand side to the first term on the right-hand side for a periodic motion of angular frequencyω:

Ratio'

mniωv⊥

qnv⊥B

' ω ωc

(1.2.26)

where ωc is the cyclotron frequency. We have taken ∂/∂t = iω and are concerned

withv⊥ only. For this drift, which is slow compared with the time scale of ωc, we

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Eq.(1.2.31). Let the electric field E and the magnetic field B be uniform, while assuming densitynand pressurephave non-zero gradients. The set of assumptions mentioned above is commonly adopted for in the magnetic confined plasma column. Please see Fig.1.2.1 for details. Taking the cross product of Eq.(1.2.25) with the magnetic field strengthB, we have

0 =qn[E×B+ (v⊥×B)×B]− ∇p×B (1.2.27)

then

0 =qnE×B−v⊥B2

[image:31.595.213.486.209.493.2]

− ∇p×B (1.2.28)

Figure 1.2.1: Fig.3.4 in [Chen, 1975], the diamagnetic drifts in a cylindrical plasma

Therefore,

v⊥=

E×B

B2 −

∇p×B

qnB2 ≡vE+vD (1.2.29)

where

vE ≡

E×B

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vD ≡ −

∇p×B

qnB2 (1.2.31)

Equations (1.2.30) and Eq.(1.2.31) define theE×B drift and diamagnetic drift respectively. The drift velocity vE is the same as for guiding centres, in

ad-dition there is a new drift velocity vD called the diamagnetic drift. Since vD is

perpendicular to the gradient direction, the neglect of (v· ∇)v is justified ifE = 0. Because under this situation, we havev =vD andvD· ∇= 0 . (v· ∇)v can still be

neglected ifE =−∇φ6= 0 and∇φ,∇pare in the same direction. BecausevD ⊥ ∇,

then (∇p×B)⊥ ∇, then (∇φ×B)⊥ ∇, thus we havevE ⊥ ∇. Using Eq.(1.2.15),

we can rewrite the diamagnetic drift velocity in Eq.(1.2.31) as

vD =± γKT

eB

ˆ

z× ∇n

n (1.2.32)

whereK is the Boltzmann’s constant and T is the temperature. In particular, for an isothermal plasma(γ = 1) in the geometry of Fig.1.2.1, where∇n=n0rˆ, we have the following formulas for a cylindrical plasma:

vDi = KTi

eB n0

n

ˆ

θ (1.2.33)

vDe = −

KTe eB

n0 n

ˆ

θ (1.2.34)

where n0 ≡ ∂n

∂r < 0[Chen, 1975]. The discussions above can be found in [Chen,

1975]. The applications of this physical mechanism will be discussed in Chapter 1.2.5 in detail.

1.2.2 Magnetohydrodynamics description

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the moments of the kinetic equations. The discussions here appear in [Li et al., 2006]. We start from the classical Maxwell’s equations,

∇ ·E = ρq

0

(1.2.35)

∇ ×E = −∂B

∂t (1.2.36)

∇ ·B = 0 (1.2.37)

∇ ×B = 0µ0∂E

∂t +µ0J (1.2.38)

where E and B are electric field and magnetic field respectively, J is the current density,ρq is density of free charge and0 andµ0 are permittivity and conductivity

in vacuum respectively. In a frame of referenceO, which is stationary with respect to fluid flow, the Ohm’s law is

J =σE (1.2.39)

and Lorenz force is

f =ρqE+J ×B (1.2.40)

We note that, in general, the Ohm’s law can take more general form, which may include electron pressure, Hall’s term, and Ohmic heating term, for example.

We now introduce another frame of reference O0, the O0 has a velocity u

with respect toO. Then in the frame of referenceO0, we have

J0 =σE0 (1.2.41)

whereJ0 andE0 are current density and electric field in O0. Then for electric field and current inO0, we have

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J0 =ρqv

0

=ρq(v−u) =J −ρqu (1.2.43)

Substituting Eqs.(1.2.42-1.2.43) into Eq.(1.2.41), we get

J =σ(E+u×B) +ρqu (1.2.44)

We use the following approximations to make simplifications:

|0∂E/∂t| |J| '

0|E/T| σ|E| =

0

σT 1 (1.2.45)

|ρqu| |J| '

|u|0|∇ ·E| |∇ ×B|/µ0

' 0µ0L T E B ' L cT 2 1 (1.2.46)

|ρqE| |J×B| '

0|∇ ·E||E| |B||∇ ×B|/µ0

'0µ0 E B 2 ' L cT 2 1 (1.2.47)

In the approximations above, we assume that the wave length λ ∼ c/ω of the field frequencyω is much larger than the characteristic lengthLof fluid motion. We also assume that the ratio of electric conductivityσto field frequencyω satisfies

σ 0ω

1, which means the characteristic time of field change is much larger than the particle collision time and the fluid is considered to be good electrical conductor. Thus the displacement current 0∂E/∂t, the current ρqu and electrical force ρqE

can be neglected.

So the Maxwell’s equations, Ohm’s law and Lorenz force equation can be transformed in an electrically conducting fluid as follows:

∇ ·E = ρq

0 (1.2.48)

∇ ×E = −∂B

∂t (1.2.49)

∇ ·B = 0 (1.2.50)

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J = σ(E+u×B) (1.2.52)

f = J×B (1.2.53)

From now on, we consider the fluid equation with electric and magnetic forces. A fluid plasma should obey the equation of continuity, which is

∂ρ

∂t +∇ ·(ρu) = 0 (1.2.54)

The motion equation of an electrically neutral and non-conducting fluid is

ρdu

dt =∇ ·P +ρg (1.2.55)

whereP is defined as tensor of stress, P = 2ηS −

p+2

3η∇ ·u−η

0

∇ ·u

I and

Sij =

1 2

∂ui ∂xj

+∂uj

∂xi

. Here we consider the additional electric and magnetic forces and omit the force due to gravity, so that the new equation is

ρdu

dt =∇ ·P +ρqE+J×B (1.2.56)

This takes following form if the fluid has no viscosity and the pressure is isotropic(η= 0, P =−pI),

ρdu

dt =−∇p+ρqE+J ×B (1.2.57)

The energy equation of a fluid is

ρd dt

+u

2

=∇ ·(P ·u) +ρg·u− ∇ ·q (1.2.58)

where q = −κ∇T, or the following form if we consider that magnitude of electric and magnetic forces is much larger than that of force due to gravity

ρd dt

+u

2

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whereρis the internal energy. Given also an equation of statep=p(ρ, T), we have the complete MHD equations as follows:

∂ρ

∂t +∇ ·(ρu) = 0 (1.2.60) ρdu

dt = ∇ ·P +ρqE+J ×B (1.2.61) ρd

dt

+u

2

= ∇ ·(P ·u) +E·J − ∇ ·q (1.2.62)

p = p(ρ, T) (1.2.63)

∇ ·E = ρq

0 (1.2.64)

∇ ×E = −∂B

∂t (1.2.65)

∇ ·B = 0 (1.2.66)

∇ ×B = µ0J (1.2.67)

J = σ(E+u×B) (1.2.68)

f = J×B (1.2.69)

where

P = 2ηS−

p+2

3η∇ ·u−η

0

∇ ·u

I (1.2.70)

Sij =

1 2

∂ui ∂xj

+∂uj

∂xi

(1.2.71)

q = −κ∇T (1.2.72)

If we assume the plasma to be an ideal conducting fluid, which means it has no viscosity, no heat conduction and is ideally conducting for current, the MHD equations transform to the ideal MHD equations which are shown here:

∂ρ

∂t +∇ ·(ρu) = 0 (1.2.73) ρdu

dt = −∇p+ρqE+J ×B (1.2.74)

pρ−γ = const. (1.2.75)

∇ ×E = −∂B

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∇ ×B = µ0J (1.2.77)

E+u×B = 0 (1.2.78)

The discussions above can be found in [Li et al., 2006].

1.2.3 L-mode

The transition from low confinement mode(L-mode) to high confinement mode(H-mode) is one of the most significant topics in fusion plasma physics. In comparison with L-mode, H-mode has higher temperature(T) and density(n) at the edge of plasma. This can contribute to a doubling of the confinement time(τ)[Wesson, 2011].

For the purpose of reaching the ignition mentioned in Chapter 1.1, it is necessary to heat the plasmas by externally additional heating, for example, neu-tral beam injection and radio-frequency heating. However, with increasing heating power, the confinement time decreases[Freidberg, 2007], please see the reviews[Doyle et al., 2007]. By analysing the experimental results from Tokamak devices, Goldston proposed the confinement scaling[Goldston, 1984],

τG= 0.037

IR1.75κ0.5

P0.5_a0.37 (1.2.79)

where τG is the confinement time in second, I is the toroidal current in MA, P is

the external heating power in MW andκ=b/a is the plasma elongation.

The Goldston’s law was found to predict the experimental data from Toka-mak quite well even before the operation of JET[Wesson, 2011]. Another time scaling called ITER89-P proposed in [Wagner et al., 1990] predicts the confinement time more precisely,

τ_EIT ER89−P = 0.048I

0.85_R1.2_a0.3_κ0.5 _n/₁₀200.1

B0.2_A0.5

P0.5 (1.2.80)

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[image:38.595.172.449.101.328.2]

Figure 1.2.2: Fig.2(a) in [Yushmanov et al., 1990], the comparisons of confinement times(τ_EIT ER89−P and τ_Eexp) in different fusion devices.

1.2.4 H-mode

In the year 1982, Wagner discovered a new confinement regime called H-mode when utilizing neutral beam heating on ASDEX in Germany. He found an increase in density caused by a sudden improvement in particle confinement[Wagner et al., 1982]. Also, the confinement time is twice that in L-mode. In the year 1984, a similar H-mode was observed in PDX Tokamak[Kaye et al., 1984], in DIII-D Tokamak in 1986[Burrell et al., 1987] and in JET Tokamak in 1987[Tanga et al., 1987b].

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In the H-mode, the change in magnetic confinement happens at the edge of plasmas, where there is an abrupt increase in the pressure gradient(∇p) associated with increasing density on the edge[Doyle et al., 2007]. The question of the H-mode formation is still open, although many scientists believe this improvement on the edge can be taken as a transport barrier[Xu et al., 2012a; Dux et al., 2014], which is a narrow region with considerably reduced transport and steep gradient by the shear effect. It generates an enhancement in stored energy, on the time scale of the confinement time[Wesson, 2011]. The reason for the formation of transport barrier is thought to be the generation of flow shear that suppresses turbulent transport in the H-mode[Diamond et al., 2005; Wesson, 2011]. For further information about the relationship between the turbulence and flow shear, please see Chapter 1.2.5. For actual applications, please see Chapters 3.2–3.3 and Chapters 4.2–4.4.

In order to realise the L-H transition, the external heating power must exceed a threshold. An empirical scaling for the threshold has been acquired from datasets of many Tokamaks[Doyle et al., 2007; Wesson, 2011], as given below

Pthr= 1.38(n/1020)0.77B0.92R1.23a0.76 (1.2.81)

where Pthr is in MW. For the comparisons of the scaling and the experimental

thresholds in different Tokamaks, please see Fig.1.2.3.

Figure 1.2.3: Fig.7 in [Doyle et al., 2007], the comparison of experimental power threshold for L-H transition with the scaling expression Eq.(1.2.81)[Wesson, 2011].

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I-phase has been observed in many Tokamaks in recent years[Conway et al., 2011; Kallenbach et al., 2011; Zhang et al., 2013; Cheng et al., 2013]. The I-phase, which is a type of limit cycle oscillation, is an intermediate phase between L-mode and H-mode. The I-phase will appear if the heating power is a little lower than the threshold for L-H transition. Sometimes the I-phase is the final states and sometimes not. It depends on the circulation direction of the limit cycle manifold, see Fig.1.2.4 for further information. The H-mode can only appear after I-phase if the circulation direction of the limit cycle(I-phase) is counter-clockwise[Cheng et al., 2013]. For the applications of L-I-H transitions and further details about the I-phase, please see Chapters 3.2–3.3 and Chapter 4.4.

Figure 1.2.4: Fig.3 in [Cheng et al., 2013], the transitions of L-I-H and L-I.

As mentioned earlier, H-mode has been found in Stellarators as well as Toka-maks. There is no toroidal current in Stellarators, see Chapter 1.3.2, which might indicate that the toroidal current plays a minor role in the transition physics[Wagner, 2007]. In ASDEX, strong isotopic effects in transport with deuterium have been dis-covered which yield a better confinement. Lower transport in deuterium plasmas allows to reach H-mode at lower external heating power[Wagner, 2007]. It is re-ported that in JET with tritium plasmas this trend still exists[Righi et al., 1999]. In addition, Stellarator plasmas do not demonstrate an isotopic effect in the con-finement. Also the power threshold, mentioned in Eq.(1.2.81), does not depend on the mass of isotopes[Wagner, 2007].

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localised modes(ELMs)-free, and the transport barrier can be produced for longer periods of time[Burrell et al., 2002, 2004]. To be explained, ELMs are significant expulsions of heat and particles with deleterious consequences for the vessel wall and machine operation[Burrell et al., 2005; Zohm, 1996], please Chapter 1.4.5. The dis-advantage of QH-mode is the impurity accumulation[Burrell et al., 2002]. QH-mode are also observed in, for instance, ASDEX Upgrade Tokamak[Suttrop et al., 2004], JET Tokamak[Suttrop et al., 2005] and W7-AS Stellarator[Hirsch et al., 2008].

1.2.5 Interactions of drift wave turbulence and zonal flow

In fluid dynamics, turbulence is defined in terms of flow regimes characterized by chaotic property changes[Bradshaw, 2013]. In plasmas, turbulence is usually con-sidered to be driven by the temperature gradient(∇T), which must also evolve in a way which is consistent with changes in the heat flux[Malkov and Diamond, 2009]. Therefore, the variation of the heat flux and its corresponding temperature gradient are both important for energy transport in fusion plasmas. At the same time, the energy transport phenomena are typically turbulent, and these are intrinsic nonlin-ear and non-diffusive[Dendy et al., 2013]. For further information in relation to the heat flux and temperature gradient, please see Chapters 2.4–2.5.

Drift waves are the most widely investigated forms of plasma turbulence in the magnetized confined plasmas[Horton et al., 2012]. The transitions from L-mode to H-mode, see Chapters 1.2.3–1.2.4, are supposed to be induced by interactions of drift wave turbulence and zonal flows[Wesson, 2011; Malkov and Diamond, 2009; Zhu et al., 2013, 2014]. A short introduction to generation of drift waves will be given as shown in [Chen, 1975].

Drift waves have a small but finite component of k along B0. Hence, the

constant density surface resembles fluted column with a slight helical twist, see Fig.1.2.5. Then we can enlarge the cross section enclosed by the box in Fig.1.2.5 and envisage it in Cartesian coordinates, see Fig.1.2.6. The driving force for this instability is temperature gradientKn∇T0 if we assume Kn=const.; or the force can be density gradientKT∇n0 if assumingKT =const.. Here we adopt the latter assumption for simplicity.

Since drift waves have a finite kz, electrons can flow along the B0 direction

to establish a thermodynamic equilibrium. Under this situation, they will obey Boltzmann’s relation in the linearised approximation,

n1 n0 =

eφ1 kTe

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[image:42.595.219.416.106.276.2]

Figure 1.2.5: Fig.6-13 in [Chen, 1975], the geometry of drift instability in a cylinder. The rectangular region is also shown in Fig.1.2.6.

In Fig.1.2.6, n1 and φ1 are both positive at point A which is in the denser

region. Based on this logic, we know that B is in the less dense region, so n1

and φ1 are negative. The difference of electric potential produces an electric field

E1 between point A and point B. This electric field E1 will cause a drift velocity v1 =E1×B0/B02 in the x-direction. As the drift waves pass by, travelling in the y-direction, oscillations of n1 and φ1 will be observed at point A. Also, the driftv1

will oscillate in time, and it isv1 that causes the oscillating density.

Since there is ∇n0 in the –x-direction, the plasmas of different density will be brought to a point A by driftv1. Although the wave travels in they-direction, a

drift wave has a motion such that the fluid moves back and forth in thex-direction. The magnitude ofv1x is

v1x = Ey B0

=−ikyφ1 B0

(1.2.83)

Then we assume that v1x does not vary with x and kz is quite small by

comparison with ky, so the fluid oscillates incompressibly in the x-direction and

therefore,

∂n1

∂t =−v1x ∂n0

∂x (1.2.84)

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[image:43.595.223.420.105.342.2]

Figure 1.2.6: Fig.6-14 in [Chen, 1975], the physical mechanism of drift waves.

drift vD. Since v1 is a transverse oscillation, that is v1 is in +/- x-direction, while

the wave vector is in they-direction, the term n0∇ ·v1 will vanish. We can rewrite

Eq.(1.2.84) by using Eq.(1.2.82) and Eq.(1.2.83),

−iωn1= ikyφ1

B0 n

0

0=−iω eφ1 KTe

n0 (1.2.85)

so that we have,

ω ky

=−KTe eB0

n0₀ n0

=vDe (1.2.86)

These waves are called drift waves that travel with the electron diamagnetic drift velocity, see Chapter 1.2.1. This drift velocity is in the y-direction, which corresponds to the azimuthal direction in cylindrical geometry[Chen, 1975]. The discussions above can be found in [Chen, 1975].

In the ideal Tokamak plasmas, the density fluctuation will cause particles to drift around the perturbation to generate a turbulent eddy[Gallagher, 2013]. This turbulent eddy will be discussed in the following paragraphs.

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or∇Ti, a relevant thermodynamic concept is the Carnot engine shown in Fig.1.2.7.

There is an input energyQ on the left side. The vortices cycle over the correlation length lc, connecting T1 and T2[Horton et al., 2012]. That means the drift wave

[image:44.595.236.406.238.387.2]

turbulence is driven by the temperature gradient[Malkov and Diamond, 2009]. For further discussions of this relationship, see [Diamond et al., 1994; Kim and Diamond, 2003; Diamond et al., 2005, 2011; Zhu et al., 2013, 2014]. See also [Tynan et al., 2009] for a review of the drift wave turbulence.

Figure 1.2.7: Fig.3.3 in [Horton et al., 2012], the diagram of the Carnot cycle for∇T

driven drift waves. W is the maximum energy released to the plasma turbulence.

As we also know, there exists a radial electric field in fusion plasmas that is often associated with the improved confinement H-mode. This radial electric field is consistent with the radial force balance equation for an ion of charge Ze(Er =

1

nZe dpi

dr −VθiBφ+VφiBθ). The radially varyingEr profile reduces the anomalous

transport, which is partly raised by drift wave turbulence in fusion plasmas[Wesson, 2011]. A shear effect produced by theE ×B force distorts the circular turbulent eddy of diameter L into an elongated elliptical shape after time t. The sheared velocitySv is superimposed on an isotropic turbulent eddy, being perpendicular to

the magnetic field[Wesson, 2011], see Fig.1.2.8 for further information. The minor axis of Fig.1.2.8(c) is reduced toL⊥ and the major axis can be calculated by L` = Lp1 +S2

vt2, see [Itoh et al., 1999]. Then the turbulent eddies can be suppressed or

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and density, wherekφand kθ are the toroidal and poloidal components of the wave

vector, respectively, such thatn and m are the corresponding mode numbers[Zhao et al., 2006]. There is also a high-frequency zonal flow called the geodesic acoustic mode(GAM). The temporal frequency of a GAM is approximately proportional to the product of the sound speed(cs) and the reciprocal of the major radius(R) of

a toroidal fusion device[Miyamoto, 2006]. Unlike low frequency zonal flow, GAM has n = 1 perturbation in density[Kr¨amer-Flecken et al., 2006]. The propagation directions of both classes of zonal flow are parallel[Zhu et al., 2014], furthermore both low and high-frequency zonal flows can suppress the drift wave turbulence intensity by shear[Itoh and Itoh, 2011]. In addition to zonal flows, mean sheared flows also play crucial roles in turbulence suppression. These mean flows are driven by the background gradients[Kim and Diamond, 2003]. Besides their shearing effects which act on turbulence, mean shear flows also affect the formation of zonal flows[Hsu and Diamond, 2015]. Please see review[Diamond et al., 2005] about zonal flows and Chapters 3–4 in this thesis for further applications to confinement regime transitions from L-mode to H-mode.

Figure 1.2.8: Fig.18.2 in [Itoh et al., 1999], the effect of a shearedE×Bflow,VE×B

on a turbulent eddy. (a) illustrates the Cartesian coordinate with magnetic field and shear velocity. (b) the circular turbulent eddy with the size L. (c) distorted turbulent eddy by sheared flow.

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di-Figure 1.2.9: Fig.1 in [Diamond et al., 2005], classic and new paradigm for plasma turbulence.

agram to show this relationship, see Fig.1.2.9. In the classic paradigm, the drift wave turbulence is driven by the free energy source and dissipated by the Landau damping. The new paradigm illustrates two relationships: the one is the relation-ship between drift wave turbulence and free energy; the other is the predator-prey relationship between zonal flows and drift waves. In Chapter 1.2.4, it was discussed how zonal flow–drift wave interactions are commonly used to explain the L–H tran-sition. Fig.1.2.9 illustrates why, in principle, we only need drift wave turbulence, zonal flow and a free energy source such as the temperature gradient to explain that phenomenon. As a result, reduced models provide an effective and interesting test of the conventional plasma physics narrative that is used to interpret the L–H transition.

We will give introductions to two representative reduced models, those of [Malkov and Diamond, 2009] and [Itoh and Itoh, 2011] in Chapter 2.4.1 and Chapter 2.4.2 respectively. For further discussion of these models, please see Chapter 3 and Chapter 4.

1.2.6 Heat pulse experiments and anomalous transport

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JT-60U[Inagaki et al., 2006] and RTP[Gorini et al., 1993; Mantica et al., 1999; Hogeweij et al., 2000]. For Stellarators, we can take LHD[Inagaki et al., 2004, 2006, 2010; Tamura et al., 2007; Dendy et al., 2013] and W7-AS[Walter et al., 1998] for examples.

Figure 1.2.10: Fig.1 in [Mantica et al., 1999], the time evolution of electron tem-peratureTe, the averaged electron density ¯ne and the electron energy stored in the

plasmas We for RTP discharge r19970224.024. A hydrogen pellet is injected at t= 0.2054 second in target plasmas.

In the experiments mentioned above, we can observe either a negative or a positive excursion in the core electron temperature from its steady state value, depending on the confinement properties of the plasmas as discussed in e.g.[Dendy et al., 2013] if we decrease the edge electron temperature by, for example, pellet injection. See Fig.1.2.10 and Fig.1.2.11 for detailed information on electron densities and electron temperatures in the RTP Tokamak and LHD Stellarator respectively.

From Fig.1.2.10, it can be seen that the trend of time series of the electron density ne and the electron temperature Te is quite different: Te returns to the

level before pellet injection while ne is still rising. Hence it is not possible to give

a phenomenological interpretation of theTe rise as due to a dependence of electron

density[Mantica et al., 1999]. However, similar phenomena are not observed in Fig.1.2.11(b) and Fig.1.2.11(e). This is an indication of the differences between heat pulse experiments in Tokamaks and Stellarators. The reason might be the distinction inq-profiles. There is largerqin outer plasmas in Tokamaks, whereas the

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For the further information about comparisons of heat pulse experiments between Tokamaks and Stellarators, please see [Inagaki et al., 2006]. It is apparent from Fig.1.2.11(a) that the transport process which underlies the heat pulse experiment is not diffusive. Consequently, the heat pulse propagation of electron temperature, which can be taken as a strong form of anomalous transport, is local and non-diffusive[Dendy et al., 2013].

Figure 1.2.11: Fig.1 in [Inagaki et al., 2010], typical electron temperatureTeresponse

to the pellet injection in (a) the local diffusive case (b) the abruptTe rise case and

(c) the abruptTe drop case.

Further heat pulse propagation experiments indicate that the core electron temperature Te rises following the edge cooling, which happens through the

for-mation of a large temperature gradient(implying a thermal transport barrier) in a radially localized region of the RTP plasmas[Mantica et al., 1999]. It is also found that the electron transport is changed in heat pulse propagation experiments in which core Te rises. That is to say, a significant decrease in electron heat

diffu-sivity χe in the thermal transport barrier region may be required to explain the

observations[Mantica et al., 1999]. Fig.5(a) in [Mantica et al., 1999] is a concrete evidence that core electron temperatureTe rise case observed in Fig.1.2.10 during

rapid edge cooling by pellet injection is accompanied by a reduction in the electron heat diffusivity in the thermal transport barrier in RTP plasmas[Mantica et al., 1999].

Similar results are obtained when analysing datasets from heat pulse propa-gation experiments on LHD. It is found that the electron heat diffusivity χe has a

nonlinear dependence on electron temperatureTe and electron temperature

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[image:49.595.178.455.109.334.2]

Figure 1.2.12: Fig.2 in [Inagaki et al., 2010], the bifurcation diagram containing stationary and dynamic state. This is expressed by the relationship between heat flux average by electron densityqe/ne and electron temperature gradient∇Te in the

core plasma(ρ= 0.19) in LHD. The green arrows denote the variation directions.

the thermal transport barrier is also found in the experiments[Inagaki et al., 2004]. Consequently a linear diffusive model, which describes diffusion phenomena in terms of qe = −neχ0(r)∇Te, cannot explain the physics of the heat pulse[Inagaki et al.,

2004]. It is also suggested that the electron heat diffusivityχ0 inside(outside) the transport barrier decreases(increases) with the increasing electron temperatureTein

LHD. Thus the properties of transport inside the barrier are qualitatively different from outside the barrier in LHD[Inagaki et al., 2004]. In conclusion, the thermal transport barrier is indeed the region in which the value of the implied electron heat diffusivity χ0 ' −∇Te/qe is reduced in the LHD Stellarator as well as in the RTP

Tokamak[Inagaki et al., 2004].

In [Inagaki et al., 2010], a bifurcation(See Chapter 2.2.3) phenomenon is found, see Fig.1.2.12 for the details. If the value of electron temperature gradient

∇Te is approximately less than 5keV/m, only local diffusive transport would be

observed. Therefore, the |∇Te| = 5keVm−1 is the threshold or bifurcating point

for diffusive/non-diffusive transport. For the branch where |∇Te| ≥ 5keVm−1, the

coreTe rise case is found if electron density averaged heat flux qe/ne is larger than

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approximately[Inagaki et al., 2010].

1.3 Magnetic confinement fusion devices

As we mentioned earlier in Chapter 1.1, magnetic confinement fusion(MCF) plas-mas should be heated up to several hundred million kelvin to achieve fusion condi-tion[McCracken and Stott, 2012]. However, there is no material which can sustain its physical properties at such a high temperature. Instead magnetic fields are used to confine the fully ionized plasmas. This leads to the concepts of Tokamaks and Stellarators, which we now review.

1.3.1 Tokamaks

The Tokamak concept was invented by physicists Andrei Sakharov and Igor Tamm. It uses magnetic field lines to confine high temperature plasmas in the shape of a torus[McCracken and Stott, 2012], see Figure 1.3.1 for a schematic diagram.

Figure 1.3.1: Schematic diagram of a Tokamak device from Wikipedia.

In order to achieve plasma equilibrium, the toroidal magnetic field Bφ is

essential[Shafranov, 1963; Mukhovatov and Shafranov, 1971]. The toroidal magnetic field line is produced by the currents in coils linking the plasma. To reach the equilibrium where the plasma pressure is balanced by the magnetic forces, it is also necessary to have a poloidal magnetic field Bp[Wesson, 2011]. In Tokamaks, this

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toroidal magnetic field and poloidal magnetic field produces helical or twisted field lines around the torus[Wesson, 2011], see the green and yellow arrows in Figure 1.3.1.

As the source of poloidal magnetic field, the plasma current[Ohkawa, 1970; Jukes, 1970] is driven by a toroidal electric field induced by a transformer in which a magnetic flux change through the torus is generated. The flux change is created by a current passed through primary coil around the torus, see Figure 1.3.2. Control of the shape needs additional toroidal currents, carried by suitable coils[Wesson, 2011]. The entire system of toroidal and poloidal coils, which together generate the Tokamak magnetic field configuration, is shown in Figure 1.3.1. For other auxiliary components of Tokamak, please see Figure 1.3.3 which shows the cross section of a Tokamak and introduction in [Freidberg, 2007] for further information.

Figure 1.3.2: Figure 1.6.3 in [Wesson, 2011]. (a) The change of flux through the torus induces toroidal electric field which drives the toroidal current. (b) The flux change is produced by primary winding using a transformer core.

In most cases, auxiliary external heating[Mirnov, 1969; Rayle et al., 1969] and current drive are essential to support nuclear fusion. There are six main optional heating methods which are ohmic heating[Artsimovitch et al., 1964], neutral beam heating[Eubank et al., 1979], radio frequency heating[Porkolab, 1977], ion cyclotron resonance heating[Perkins, 1977], lower hybrid resonance heating[Perkins, 1977] and electron cyclotron resonance heating[Ott et al., 1980]. For their applications, please see Chapters 1.2.3–1.2.4 for transitions from low confinement regime(L-mode) to high confinement regime(H-mode), and see Chapters 3 and 4 for the consequences of different levels of external heating flux for predator-prey models. These include period doubling bifurcation and chaotic behaviour of the confinement properties.